Simple forcing notions and forcing axioms

Simple Forcing Notions and Forcing Axioms

The Proper and Semi-proper Forcing Axioms for Forcing Notions That Preserve

We prove that the PFA lottery preparation of a strongly unfoldable cardinal κ under ¬0 forces PFA(א2-preserving), PFA(א3-preserving) and PFAא2 , with 2 ω = κ = א2. The method adapts to semi-proper forcing, giving SPFA(א2-preserving), SPFA(א3-preserving) and SPFAא2 from the same hypothesis. It follows by a result of Miyamoto that the existence of a strongly unfoldable cardinal is equiconsistent ...

Forcing Axioms and Cardinal Arithmetic

We survey some recent results on the impact of strong forcing axioms such as the Proper Forcing Axiom PFA and Martin’s Maximum MM on cardinal arithmetic. We concentrate on three combinatorial principles which follow from strong forcing axioms: stationary set reflection, Moore’s Mapping Reflection Principle MRP and the P-ideal dichotomy introduced by Abraham and Todorčević which play the key rol...

Universal forcing notions and ideals

The main result of this paper is a partial answer to [6, Problem 5.5]: a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the σ–ideals determined by those universality parameters. One of the most striking differences between measure and catego...

Hierarchies of forcing axioms I

We prove new upper bound theorems on the consistency strengths of SPFA(θ), SPFA(θ-linked) and SPFA(θ-cc). Our results are in terms of (θ,Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(c-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper....